# Sobol’ analysis¶

import pprint

from matplotlib import pyplot as plt
from numpy import pi

from gemseo.algos.parameter_space import ParameterSpace
from gemseo.api import create_discipline
from gemseo.uncertainty.sensitivity.sobol.analysis import SobolAnalysis


In this example, we consider a function from $$[-\pi,\pi]^3$$ to $$\mathbb{R}^3$$:

$(y_1,y_2)=\left(f(x_1,x_2,x_3),f(x_2,x_1,x_3)\right)$

where $$f(a,b,c)=\sin(a)+7\sin(b)^2+0.1*c^4\sin(a)$$ is the Ishigami function:

expressions = {
"y1": "sin(x1)+7*sin(x2)**2+0.1*x3**4*sin(x1)",
"y2": "sin(x2)+7*sin(x1)**2+0.1*x3**4*sin(x2)",
}
discipline = create_discipline(
"AnalyticDiscipline", expressions_dict=expressions, name="Ishigami2"
)


Then, we consider the case where the deterministic variables $$x_1$$, $$x_2$$ and $$x_3$$ are replaced with the uncertain variables $$X_1$$, $$X_2$$ and $$X_3$$. The latter are independent and identically distributed according to an uniform distribution between $$-\pi$$ and $$\pi$$:

space = ParameterSpace()
for variable in ["x1", "x2", "x3"]:
variable, "OTUniformDistribution", minimum=-pi, maximum=pi
)


From that, we would like to carry out a sensitivity analysis with the random outputs $$Y_1=f(X_1,X_2,X_3)$$ and $$Y_2=f(X_2,X_1,X_3)$$. For that, we can compute the correlation coefficients from a SobolAnalysis:

sobol = SobolAnalysis(discipline, space, 100)
sobol.main_method = "total"
sobol.compute_indices()


Out:

{'first': {'y1': [{'x1': array([0.19426511]), 'x2': array([0.21211854]), 'x3': array([0.01782884])}], 'y2': [{'x1': array([0.75387699]), 'x2': array([0.26516667]), 'x3': array([0.35758815])}]}, 'total': {'y1': [{'x1': array([0.87052721]), 'x2': array([0.37726189]), 'x3': array([0.29945874])}], 'y2': [{'x1': array([0.36991381]), 'x2': array([0.60974968]), 'x3': array([0.34776012])}]}}


The resulting indices are the first and total order Sobol’ indices:

pprint.pprint(sobol.indices)


Out:

{'first': {'y1': [{'x1': array([0.19426511]),
'x2': array([0.21211854]),
'x3': array([0.01782884])}],
'y2': [{'x1': array([0.75387699]),
'x2': array([0.26516667]),
'x3': array([0.35758815])}]},
'total': {'y1': [{'x1': array([0.87052721]),
'x2': array([0.37726189]),
'x3': array([0.29945874])}],
'y2': [{'x1': array([0.36991381]),
'x2': array([0.60974968]),
'x3': array([0.34776012])}]}}


They can also be accessed separately:

pprint.pprint(sobol.first_order_indices)
pprint.pprint(sobol.total_order_indices)


Out:

{'y1': [{'x1': array([0.19426511]),
'x2': array([0.21211854]),
'x3': array([0.01782884])}],
'y2': [{'x1': array([0.75387699]),
'x2': array([0.26516667]),
'x3': array([0.35758815])}]}
{'y1': [{'x1': array([0.87052721]),
'x2': array([0.37726189]),
'x3': array([0.29945874])}],
'y2': [{'x1': array([0.36991381]),
'x2': array([0.60974968]),
'x3': array([0.34776012])}]}


The main indices corresponds to the Spearman correlation indices (this main method can be changed with SobolAnalysis.main_method):

pprint.pprint(sobol.main_indices)

pprint.pprint(sobol.get_intervals())


Out:

{'y1': [{'x1': array([0.87052721]),
'x2': array([0.37726189]),
'x3': array([0.29945874])}],
'y2': [{'x1': array([0.36991381]),
'x2': array([0.60974968]),
'x3': array([0.34776012])}]}
{'y1': [{'x1': array([-0.05887176,  0.57198378]),
'x2': array([0.0281138 , 0.66092945]),
'x3': array([-0.48592145,  0.41401005])}],
'y2': [{'x1': array([0.15165848, 2.21224422]),
'x2': array([-0.24741228,  1.07690768]),
'x3': array([-0.40957101,  2.01204153])}]}


We can also sort the input parameters by decreasing order of influence: and observe that this ranking is not the same for both outputs:

print(sobol.sort_parameters("y1"))
print(sobol.sort_parameters("y2"))


Out:

['x1', 'x2', 'x3']
['x2', 'x1', 'x3']


Lastly, we can use the method SobolAnalysis.plot() to visualize both first and total order Sobol’ indices:

sobol.plot("y1", save=False, show=False)
sobol.plot("y2", save=False, show=False)
# Workaround for HTML rendering, instead of show=True
plt.show()


Total running time of the script: ( 0 minutes 0.614 seconds)

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